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The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid

Vasylkevych, Sergiy and Marsden, Jerrold E. (2005) The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid. Dynamics of Partial Differential Equations, 2 (4). pp. 281-300. ISSN 1548-159X.

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This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n=2) + 1). The idea of how this diculty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure.

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Additional Information:© 2005 International Press. Communicated by Tudor Ratiu, received August 23, 2005. The hardcopy and electronic editions of Dynamics of Partial Differential Equations are protected by the copyright of International Press.
Subject Keywords:Euler equations, Poisson map, Lie-Poisson bracket, Lagrangian representation, Lie-Poisson reduction procedure.
Classification Code:1991 MSC: Primary: 35; Secondary: 76.
Record Number:CaltechAUTHORS:20100917-074331134
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19996
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 21:17
Last Modified:26 Dec 2012 12:26

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